$G$ is a Lie group, $\mathfrak{g}$ its Lie algebra and $ad_X$ the adjoint action at $X\in \mathfrak{g}$.
I would like to know on what class of Lie groups all the $ad_X$ are diagonalizable. A similar question was already posted here. The accepted answer provides an example where $ad_X$ is not diagonalisable and mentions the Jordan-Chevalley decomposition (I am not sure I understand its implications).
Questions:
-do such groups correspond to a well known class of Lie groups? (I gess not given the anwser of the post)
-what are the standard classes of Lie groups on which we diagonalize $ad_X$, for all $X$?
Thanks